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1 This article was downloaded by: [University of Miami] On: 27 November 2012, At: 08:47 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Communications in Algebra Publication details, including instructions for authors and subscription information: A PARTIAL ORDER ON THE ORTHOGONAL GROUP Thomas Brady a & Colum Watt b a School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin, Ireland b School of Mathematics, Trinity College, Dublin, Ireland Version of record first published: 01 Sep To cite this article: Thomas Brady & Colum Watt (2002): A PARTIAL ORDER ON THE ORTHOGONAL GROUP, Communications in Algebra, 30:8, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 MARCEL DEKKER, INC. 270 MADISON AVENUE NEW YORK, NY Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. COMMUNICATIONS IN ALGEBRA Vol. 30, No. 8, pp , 2002 A PARTIAL ORDER ON THE ORTHOGONAL GROUP Thomas Brady 1 and Colum Watt 2 1 School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland tom.brady@dcu.ie 2 School of Mathematics, Trinity College, Dublin 2, Ireland colum@maths.tcd.ie ABSTRACT We define a natural partial order on the orthogonal group and completely describe the intervals in this partial order. The main technical ingredient is that an orthogonal transformation induces a unique orthogonal transformation on each subspace of the orthogonal complement of its fixed subspace. Let V be an n-dimensional vector space over a field F and let OðV Þ be the orthogonal group of V with respect to a fixed anisotropic symmetric bilinear form h ; i. In this note we will define a natural partial order on OðV Þ and completely describe the intervals in this partial order. The main technical ingredient is that an orthogonal transformation A on V induces a unique orthogonal transformation on each subspace of the orthogonal complement of the fixed subspace of A DOI: /AGB Copyright # 2002 by Marcel Dekker, Inc (Print); (Online)

3 3750 BRADY AND WATT Recall that A 2 OðV Þ if A : V! V is linear and satisfies hað~vþ; Að~wÞi ¼ h~v; ~wi for all ~v; ~w 2 V. For standard results on symmetric bilinear forms and their associated orthogonal groups see [1], but note that we are making the further assumption that the form is anisotropic. For each A 2 OðV Þ, we define two subspaces of V, FðAÞ ¼kerðA IÞ and MðAÞ ¼imðA IÞ, where I is the identity operator on V. We note that FðAÞ is the þ1-eigenspace of A, sometimes called the fixed subspace of A.We will write V ¼ V 1? V 2 whenever V is the orthogonal direct sum of subspaces V 1 and V 2. Proposition 1. V ¼ FðAÞ?MðAÞ Proof. Since the dimensions of FðAÞ and MðAÞ are complementary and the form is anisotropic, it suffices to show that these subspaces are orthogonal. So let ~x 2 FðAÞ and ~y 2 MðAÞ. Then ~x ¼ Að~xÞ and ~y ¼ðA IÞ~z for some ~z 2 V. Thus h~x;~yi ¼h~x; ða IÞ~zi ¼h~x; Að~zÞi h~x;~zi ¼hAð~xÞ; Að~zÞi h~x;~zi ¼0: We will be concerned with how the dimensions of these subspaces behave when we take products in OðV Þ. For notational convenience we will write juj for dimðuþ. Proposition 2. jmðabþj jmðaþj þ jmðbþj for A; B 2 OðV Þ. Proof. Using the identities juj þjvj ¼jUþVj þju \ Vj, FðAÞ\ FðBÞ FðABÞ and FðAÞþFðBÞ V we find that jfðaþj þ jfðbþj ¼ jfðaþþfðbþj þ jfðaþ\fðbþj n þjfðabþj; from which the result follows. This result is proved in a more general setting in. [2] However, from the proof above we see that equality occurs if and only if FðAÞ\FðBÞ ¼FðABÞ and FðAÞþFðBÞ ¼V: Thus, using the identities ½U þ VŠ? ¼ U? \ V? and U? þ V? ¼½U \ VŠ? we get the following characterization. Corollary 1. jmðabþj ¼ jmðaþj þ jmðbþj, MðABÞ ¼MðAÞMðBÞ. Definition 1. We will write A C if jmðcþj ¼ jmðaþj þ jmða 1 CÞj. Proposition 3. The relation is a partial order on OðV Þ and satisfies A B C ) A 1 B A 1 C:

4 PARTIAL ORDER ON ORTHOGONAL GROUP 3751 Proof. Reflexivity is immediate. To establish antisymmetry suppose A C and C A. Then jmðcþj ¼ jmðaþj þ jmða 1 CÞj ¼ jmðcþj þ jmðc 1 AÞj þ jmða 1 CÞj giving FðC 1 AÞ¼FðA 1 CÞ¼V or A ¼ C. To establish transitivity, suppose A B and B C. Then jmðcþj jmðaþj þ jmða 1 CÞj ¼jMðAÞj þ jmða 1 BB 1 CÞj jmðaþj þ jmða 1 BÞj þ jmðb 1 CÞj ¼jMðAÞj þ fjmðbþj jmðaþjg þ fjmðcþj jmðbþjg ¼jMðCÞj So both of the inequalities are actually equalities. The first line gives A C and is transitive. The third line gives the second assertion above. The association of the subspace MðAÞ to an element A 2 OðV Þ defines a map M from OðV Þ to the set of subspaces of V. The next sequence of lemmas shows that the restriction of M to the interval ½I; CŠ ¼fA 2 OðV Þj A Cg is a bijection onto the set of subspaces of MðCÞ. In what follows we fix C and a subspace W of MðCÞ and we suppose that A 2 OðV Þ satisfies MðAÞ ¼W. We define U to be the unique subspace of MðCÞ which satisfies juj ¼jWj and ðc IÞU ¼ W. This is possible since C I is invertible when restricted to MðCÞ. Lemma 1. If W MðCÞ then V ¼ W? U. Proof. Since the subspaces have complementary dimensions it suffices to show that their intersection is trivial. So let ~x 2 W? \ U. Then ~x 2 W? and ðc IÞ~x ¼ ~w for some ~w 2 W. Thus C~x ¼ ~x þ ~w, with ~x 2 W? and ~w 2 W so that h~x;~xi ¼ hc~x; C~xi ¼ h~x þ ~w;~x þ ~wi ¼ h~x;~xi þ h~w; ~wi: Thus ~w ¼ ~ 0 since h ; i is anisotropic and ~x ¼ ~ 0 since C I is an isomorphism on MðCÞ. Lemma 2. FðA 1 CÞFðCÞ?U. Proof. Let ~x 2 FðA 1 CÞ. Then A 1 C~x ¼ ~x, which implies C~x ¼ A~x and ðc IÞ~x ¼ðA IÞ~x. Using V ¼ FðCÞ?MðCÞ we can express ~x uniquely as ~x ¼ ~y þ ~z with ~y 2 FðCÞ and ~z 2 MðCÞ. Thus

5 3752 BRADY AND WATT ðc IÞ~z ¼ðC IÞ~x ¼ðA IÞ~x 2 MðAÞ ¼W; giving ~z 2 U. This gives FðA 1 CÞFðCÞþU and the orthogonality of the subspaces follows since U MðCÞ. Lemma 3. If MðAÞ ¼W and A C then FðA 1 CÞ¼FðCÞ?U. Proof. Since A C we have jmða 1 CÞj ¼ jmðcþj jmðaþj so that jfða 1 CÞj ¼ n jmða 1 CÞj ¼ n jmðcþj þ jwj ¼jFðCÞj þ juj: This dimension calculation can now be combined with Lemma 2. It is now possible to give a formula for A. IfV ¼ V 1 V 2 we define the projection Proj V 2 V 1 to be the linear transformation which coincides with the identity on V 1 and with the zero transformation on V 2. Lemma 4. If A C and MðAÞ ¼W then A ¼ I þðc IÞProj W? U. Proof. If MðAÞ ¼W then FðAÞ ¼W? so that A coincides with I on W?. Since FðA 1 CÞ contains U by Lemma 3, A coincides with C on U. Thus A I coincides with the zero transformation on W? and with C I on U, giving A I ¼ðC IÞProj W? U, by Lemma 1. It is not at all clear from this formula that A is orthogonal. However this is indeed the case. Lemma 5. A ¼ I þðc IÞProj W? U 2 OðV Þ. Proof. Let ~x;~y 2 V and use Lemma 1 to express ~x ¼ ~x 1 þ ~x 2, ~y ¼ ~y 1 þ ~y 2, with ~x 1 ;~y 1 2 U and ~x 2 ;~y 2 2 W?. Then, using the fact that A coincides with I on W? and with C on U, hað~xþ; Að~yÞi ¼ hcð~x 1 Þþ~x 2 ; Cð~y 1 Þþ~y 2 i ¼hC~x 1 ; C~y 1 iþhc~x 1 ;~y 2 iþh~x 2 ; C~y 1 iþh~x 2 ;~y 2 i ¼h~x 1 ;~y 1 iþhc~x 1 ;~y 2 iþh~x 2 ; C~y 1 iþh~x 2 ;~y 2 i ¼h~x;~yiþhðC IÞ~x 1 ;~y 2 iþh~x 2 ; ðc IÞ~y 1 i ¼h~x;~yi; since both ðc IÞ~x 1 and ðc IÞ~y 1 lie in W. We will call A the transformation induced by C on W. Combining the above lemmas we get the following result. Theorem 1. If C 2 OðV Þ and W is a subspace of MðCÞ then there exists a unique A 2 OðV Þ satisfying A C and MðAÞ ¼W.

6 PARTIAL ORDER ON ORTHOGONAL GROUP 3753 The induced transformations are familiar objects for two special classes of subspace. Corollary 2. If W MðCÞ is an invariant subspace of C, then the induced transformation on W is the restriction of C to W. Proof. In this case, U ¼ W and the projection in the formula for A becomes an orthogonal projection. Corollary 3. If charðfþ 6¼ 2 and W is a one dimensional subspace of MðCÞ then the orthogonal transformation induced by C on W is always the orthogonal reflection in W?. Proof. Since W is one-dimensional A must act on W by multiplication by a scalar a. The orthogonality of A forces a 2 ¼ 1 and W ¼ MðAÞ gives a 6¼ 1. The poset ðoðv Þ; Þ is not a lattice, since distinct elements C 1 and C 2 with MðC 1 Þ¼MðC 2 Þ cannot have a common upper bound. However the intervals are lattices and can be easily described. Theorem 2. If A C in OðV Þ and jmðcþj jmðaþj ¼ m then the interval ½A; CŠ ¼fB 2 OðV ÞjA B Cg is isomorphic to the lattice of subspaces of F m under inclusion. Proof. The lattices of subspaces of F m under inclusion is isomorphic to the interval ½MðAÞ; MðCÞŠ in the lattice of subspaces of V. The function B 7! MðBÞ is a bijection from the interval ½A; CŠ to the latter interval by Theorem 1. This map respects the partial orders by Corollary 1. To see that the inverse map respects the partial orders suppose that MðAÞ W 1 W 2 MðCÞ. Let B 1, B 2 be the transformations induced on W 1, W 2 respectively by C and let B 0 1 be the transformation induced on W 1 by B 2. Then B 0 1 B 2 C gives B 0 1 C, but MðB 1Þ¼MðB 0 1 Þ¼W 1 so the uniqueness part of Theorem 1 gives B 1 ¼ B 0 1 and B 1 B 2. Each chain in MðCÞ thus gives rise to a special factorization of C. Corollary 4. If C 2 OðV Þ and W 1 W 2 W k ¼ MðCÞ is a chain of subspaces in MðCÞ then C factors uniquely as a product of k transformations C ¼ B 1 B 2 ;...; B k, with B 1 B 2 ;...; B i C and MðB 1 B 2 ;...; B i Þ¼W i. Proof. If we define C i to be the transformation induced by C on W i then B i ¼ðC i 1 Þ 1 C i. The case where this chain is maximal gives a strong version of the Cartan-Dieudonne theorem.

7 3754 BRADY AND WATT Corollary 5. If charðfþ 6¼ 2, C 2 OðV Þ with jmðcþj ¼ k and W 1 W 2 W k ¼ MðCÞ is a maximal flag in MðCÞ then C factors uniquely as a product of k reflections, C ¼ R 1 R 2 R k, with MðR 1 R 2 R i Þ¼W i. Proof. Here the transformation B i defined in Corollary 4 satisfies jmðb i Þj ¼ 1 so that B i is a reflection by Corollary 3. Note 1. Using similar methods one can prove analogs of all the above results in the case of a unitary transformation over a finite-dimensional complex vector space. In this case we deal with complex linear subspaces, the induced transformations are unitary (and hence complex linear) and complex reflections replace the above reflections. REFERENCES 1. Jacobson N., Basic Algebra I, Second Edn.; Freeman: San Francisco, Scherk P., On the Decomposition of Orthogonalities into Symmetries, Proc. Amer. Math. Soc. 1950, 1, Received March 2001